Spectrum of tridiagonal block matrix The Next CEO of Stack OverflowFinding eigenvalues of a...
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Spectrum of tridiagonal block matrix
The Next CEO of Stack OverflowFinding eigenvalues of a block matrixDeterminant of block tridiagonal Toeplitz matricesEigenvalues of a block diagonal matrixIvertibility , positive definiteness of block tridiagonal matrix which arose from poisson 2-d discretizationSymmetric block matrix relatedEigenvalues of block matrix of order $m+1$About the eigenvalues of a block Toeplitz (tridiagonal) matrixEigenvalues and Eigenvectors of a Tridiagonal Block Toeplitz MatrixEigenvalues and Eigenvectors of a block tridiagonal block MatrixEigenvalues and eigenvectors of a Block Tridiagonal Matrix
$begingroup$
I have the following $4n times 4n$ block tridiagonal matrix:
$$
begin{bmatrix}
M_{1} && -M_{2} && 0 && cdots&&&& &&0 \
M_{2} && M_{1} && -JI_{2} &&cdots&&&&&& 0\
0&& JI_{2} && 0 && -hI_{2} &&&& && vdots\
0&& 0 && hI_{2} && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_{2} && 0\
0 && && &&&&JI_{2}&& M_{1}^{'}&&-M_{2}^{'}\
0 && 0 &&&& cdots && 0&&M_{2}^{'} &&M_{1}^{'}
end{bmatrix}
$$
Each block is a $2times2$ matrix with complex entries and $I_{2}$ is the $2times2$ identity matrix and $0neq h,J in mathbb{R}$ and I know the following:
- $M_{1},M_{1}^{'}$ are diagonal matrices
- $M_{2}$ and $M_{2}^{'}$ are of the form $begin{bmatrix} h && * \ 0 && hend{bmatrix}$
My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?
linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
$endgroup$
|
show 1 more comment
$begingroup$
I have the following $4n times 4n$ block tridiagonal matrix:
$$
begin{bmatrix}
M_{1} && -M_{2} && 0 && cdots&&&& &&0 \
M_{2} && M_{1} && -JI_{2} &&cdots&&&&&& 0\
0&& JI_{2} && 0 && -hI_{2} &&&& && vdots\
0&& 0 && hI_{2} && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_{2} && 0\
0 && && &&&&JI_{2}&& M_{1}^{'}&&-M_{2}^{'}\
0 && 0 &&&& cdots && 0&&M_{2}^{'} &&M_{1}^{'}
end{bmatrix}
$$
Each block is a $2times2$ matrix with complex entries and $I_{2}$ is the $2times2$ identity matrix and $0neq h,J in mathbb{R}$ and I know the following:
- $M_{1},M_{1}^{'}$ are diagonal matrices
- $M_{2}$ and $M_{2}^{'}$ are of the form $begin{bmatrix} h && * \ 0 && hend{bmatrix}$
My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?
linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
$endgroup$
$begingroup$
What is the matrix $J$ and what is $h$?
$endgroup$
– JimmyK4542
Feb 25 '18 at 3:34
$begingroup$
sorry, they are real numbers. made the change
$endgroup$
– user1058860
Feb 25 '18 at 3:39
$begingroup$
Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
$endgroup$
– Chickenmancer
Feb 25 '18 at 4:11
$begingroup$
where did you get that from?
$endgroup$
– user1058860
Feb 25 '18 at 4:15
$begingroup$
Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
$endgroup$
– JimmyK4542
Feb 25 '18 at 4:58
|
show 1 more comment
$begingroup$
I have the following $4n times 4n$ block tridiagonal matrix:
$$
begin{bmatrix}
M_{1} && -M_{2} && 0 && cdots&&&& &&0 \
M_{2} && M_{1} && -JI_{2} &&cdots&&&&&& 0\
0&& JI_{2} && 0 && -hI_{2} &&&& && vdots\
0&& 0 && hI_{2} && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_{2} && 0\
0 && && &&&&JI_{2}&& M_{1}^{'}&&-M_{2}^{'}\
0 && 0 &&&& cdots && 0&&M_{2}^{'} &&M_{1}^{'}
end{bmatrix}
$$
Each block is a $2times2$ matrix with complex entries and $I_{2}$ is the $2times2$ identity matrix and $0neq h,J in mathbb{R}$ and I know the following:
- $M_{1},M_{1}^{'}$ are diagonal matrices
- $M_{2}$ and $M_{2}^{'}$ are of the form $begin{bmatrix} h && * \ 0 && hend{bmatrix}$
My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?
linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
$endgroup$
I have the following $4n times 4n$ block tridiagonal matrix:
$$
begin{bmatrix}
M_{1} && -M_{2} && 0 && cdots&&&& &&0 \
M_{2} && M_{1} && -JI_{2} &&cdots&&&&&& 0\
0&& JI_{2} && 0 && -hI_{2} &&&& && vdots\
0&& 0 && hI_{2} && 0 &&ddots&& &&\
vdots && &&&& ddots&&ddots && -JI_{2} && 0\
0 && && &&&&JI_{2}&& M_{1}^{'}&&-M_{2}^{'}\
0 && 0 &&&& cdots && 0&&M_{2}^{'} &&M_{1}^{'}
end{bmatrix}
$$
Each block is a $2times2$ matrix with complex entries and $I_{2}$ is the $2times2$ identity matrix and $0neq h,J in mathbb{R}$ and I know the following:
- $M_{1},M_{1}^{'}$ are diagonal matrices
- $M_{2}$ and $M_{2}^{'}$ are of the form $begin{bmatrix} h && * \ 0 && hend{bmatrix}$
My question is: Can we compute the eigenvalues and eigenvectors exactly knowing this information?
linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices
linear-algebra matrices eigenvalues-eigenvectors block-matrices tridiagonal-matrices
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
edited Mar 16 at 8:06
Rodrigo de Azevedo
13.2k41960
13.2k41960
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
asked Feb 24 '18 at 22:22
user1058860user1058860
285111
285111
$begingroup$
What is the matrix $J$ and what is $h$?
$endgroup$
– JimmyK4542
Feb 25 '18 at 3:34
$begingroup$
sorry, they are real numbers. made the change
$endgroup$
– user1058860
Feb 25 '18 at 3:39
$begingroup$
Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
$endgroup$
– Chickenmancer
Feb 25 '18 at 4:11
$begingroup$
where did you get that from?
$endgroup$
– user1058860
Feb 25 '18 at 4:15
$begingroup$
Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
$endgroup$
– JimmyK4542
Feb 25 '18 at 4:58
|
show 1 more comment
$begingroup$
What is the matrix $J$ and what is $h$?
$endgroup$
– JimmyK4542
Feb 25 '18 at 3:34
$begingroup$
sorry, they are real numbers. made the change
$endgroup$
– user1058860
Feb 25 '18 at 3:39
$begingroup$
Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
$endgroup$
– Chickenmancer
Feb 25 '18 at 4:11
$begingroup$
where did you get that from?
$endgroup$
– user1058860
Feb 25 '18 at 4:15
$begingroup$
Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
$endgroup$
– JimmyK4542
Feb 25 '18 at 4:58
$begingroup$
What is the matrix $J$ and what is $h$?
$endgroup$
– JimmyK4542
Feb 25 '18 at 3:34
$begingroup$
What is the matrix $J$ and what is $h$?
$endgroup$
– JimmyK4542
Feb 25 '18 at 3:34
$begingroup$
sorry, they are real numbers. made the change
$endgroup$
– user1058860
Feb 25 '18 at 3:39
$begingroup$
sorry, they are real numbers. made the change
$endgroup$
– user1058860
Feb 25 '18 at 3:39
$begingroup$
Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
$endgroup$
– Chickenmancer
Feb 25 '18 at 4:11
$begingroup$
Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
$endgroup$
– Chickenmancer
Feb 25 '18 at 4:11
$begingroup$
where did you get that from?
$endgroup$
– user1058860
Feb 25 '18 at 4:15
$begingroup$
where did you get that from?
$endgroup$
– user1058860
Feb 25 '18 at 4:15
$begingroup$
Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
$endgroup$
– JimmyK4542
Feb 25 '18 at 4:58
$begingroup$
Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
$endgroup$
– JimmyK4542
Feb 25 '18 at 4:58
|
show 1 more comment
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$begingroup$
What is the matrix $J$ and what is $h$?
$endgroup$
– JimmyK4542
Feb 25 '18 at 3:34
$begingroup$
sorry, they are real numbers. made the change
$endgroup$
– user1058860
Feb 25 '18 at 3:39
$begingroup$
Its eigenvalues are real . . . en.wikipedia.org/wiki/Tridiagonal_matrix
$endgroup$
– Chickenmancer
Feb 25 '18 at 4:11
$begingroup$
where did you get that from?
$endgroup$
– user1058860
Feb 25 '18 at 4:15
$begingroup$
Also, what is the pattern of $JI_2$ and $hI_2$ blocks? Is it one $JI_2$ block, followed by $4n-5$ $hI_2$ blocks, followed by one $JI_2$ block, or is the pattern something else?
$endgroup$
– JimmyK4542
Feb 25 '18 at 4:58